• Home
• /
• Blog
• /
• Rationalize The Denominator(With Examples)

Rationalize The Denominator(With Examples)

August 17, 2022

This post is also available in: हिन्दी (Hindi)

Sometimes during calculation, you come across fractions having irrational numbers as their numerators or denominators. As it’s a convention to have the denominator of a fraction as a rational number, therefore, in cases where the denominator of a fraction is an irrational number, you have to make sure that its denominator should have a rational number.

The procedure of converting the irrational denominator into a rational denominator is called rationalizing the denominator. The expression or a number that is used to convert an irrational denominator into a rational denominator is called a rationalizing factor.

What is Rationalizing?

In mathematics, rationalizing means converting an irrational number or an irrational expression into a rational number or a rational expression. The process of conversion from an irrational value to a rational value is done by multiplying an irrational number or an expression by another irrational number or an irrational expression. The irrational number or an irrational expression that is multiplied is called the rationalizing factor (RF).

For example, to rationalize $\sqrt{2}$, you need another $\sqrt{2}$ so that you get $\sqrt{2} \times \sqrt{2} = 2$.

Note: $\sqrt{2}$ is an irrational number and $2$ is a rational number.

Therefore, the rationalizing factor of $\sqrt{2}$ is $\sqrt{2}$.

Similarly, to rationalize $\sqrt[3]{3}$, you need another number $\sqrt[3]{9}$ so that you get $\sqrt[3]{3} \times \sqrt[3]{9} = \sqrt[3]{27} = 3$.

Here $\sqrt[3]{3}$ is an irrational number whereas $3$ is a rational number.

What Does Rationalize The Denominator Mean?

Rationalizing the denominator means the process of removing an irrational number from the denominator of a fraction. The convention is that the denominator of a fraction should always be a rational number. Further by doing so, we bring the fraction to its simplest form.

For example, $\frac {1}{\sqrt {2}}$ after rationalizing the denominator becomes $\frac {\sqrt{2}}{2}$.

Similarly, $\frac {1}{\sqrt[3]{2}}$ after rationaling the denominator becomes $\frac {\sqrt[3]{4}}{2}$.

Is your child struggling with Maths?
We can help!
Country
• Afghanistan 93
• Albania 355
• Algeria 213
• American Samoa 1-684
• Andorra 376
• Angola 244
• Anguilla 1-264
• Antarctica 672
• Antigua & Barbuda 1-268
• Argentina 54
• Armenia 374
• Aruba 297
• Australia 61
• Austria 43
• Azerbaijan 994
• Bahamas 1-242
• Bahrain 973
• Belarus 375
• Belgium 32
• Belize 501
• Benin 229
• Bermuda 1-441
• Bhutan 975
• Bolivia 591
• Bosnia 387
• Botswana 267
• Bouvet Island 47
• Brazil 55
• British Indian Ocean Territory 246
• British Virgin Islands 1-284
• Brunei 673
• Bulgaria 359
• Burkina Faso 226
• Burundi 257
• Cambodia 855
• Cameroon 237
• Cape Verde 238
• Caribbean Netherlands 599
• Cayman Islands 1-345
• Central African Republic 236
• Chile 56
• China 86
• Christmas Island 61
• Cocos (Keeling) Islands 61
• Colombia 57
• Comoros 269
• Congo - Brazzaville 242
• Congo - Kinshasa 243
• Cook Islands 682
• Costa Rica 506
• Croatia 385
• Cuba 53
• Cyprus 357
• Czech Republic 420
• Denmark 45
• Djibouti 253
• Dominica 1-767
• Egypt 20
• Equatorial Guinea 240
• Eritrea 291
• Estonia 372
• Ethiopia 251
• Falkland Islands 500
• Faroe Islands 298
• Fiji 679
• Finland 358
• France 33
• French Guiana 594
• French Polynesia 689
• French Southern Territories 262
• Gabon 241
• Gambia 220
• Georgia 995
• Germany 49
• Ghana 233
• Gibraltar 350
• Greece 30
• Greenland 299
• Guam 1-671
• Guatemala 502
• Guernsey 44
• Guinea 224
• Guinea-Bissau 245
• Guyana 592
• Haiti 509
• Heard & McDonald Islands 672
• Honduras 504
• Hong Kong 852
• Hungary 36
• Iceland 354
• India 91
• Indonesia 62
• Iran 98
• Iraq 964
• Ireland 353
• Isle of Man 44
• Israel 972
• Italy 39
• Jamaica 1-876
• Japan 81
• Jersey 44
• Jordan 962
• Kazakhstan 7
• Kenya 254
• Kiribati 686
• Kuwait 965
• Kyrgyzstan 996
• Laos 856
• Latvia 371
• Lebanon 961
• Lesotho 266
• Liberia 231
• Libya 218
• Liechtenstein 423
• Lithuania 370
• Luxembourg 352
• Macau 853
• Macedonia 389
• Malawi 265
• Malaysia 60
• Maldives 960
• Mali 223
• Malta 356
• Marshall Islands 692
• Martinique 596
• Mauritania 222
• Mauritius 230
• Mayotte 262
• Mexico 52
• Micronesia 691
• Moldova 373
• Monaco 377
• Mongolia 976
• Montenegro 382
• Montserrat 1-664
• Morocco 212
• Mozambique 258
• Myanmar 95
• Namibia 264
• Nauru 674
• Nepal 977
• Netherlands 31
• New Caledonia 687
• New Zealand 64
• Nicaragua 505
• Niger 227
• Nigeria 234
• Niue 683
• Norfolk Island 672
• North Korea 850
• Northern Mariana Islands 1-670
• Norway 47
• Oman 968
• Pakistan 92
• Palau 680
• Palestine 970
• Panama 507
• Papua New Guinea 675
• Paraguay 595
• Peru 51
• Philippines 63
• Pitcairn Islands 870
• Poland 48
• Portugal 351
• Puerto Rico 1
• Qatar 974
• Romania 40
• Russia 7
• Rwanda 250
• Samoa 685
• San Marino 378
• Saudi Arabia 966
• Senegal 221
• Serbia 381 p
• Seychelles 248
• Sierra Leone 232
• Singapore 65
• Slovakia 421
• Slovenia 386
• Solomon Islands 677
• Somalia 252
• South Africa 27
• South Georgia & South Sandwich Islands 500
• South Korea 82
• South Sudan 211
• Spain 34
• Sri Lanka 94
• Sudan 249
• Suriname 597
• Svalbard & Jan Mayen 47
• Swaziland 268
• Sweden 46
• Switzerland 41
• Syria 963
• Sao Tome and Principe 239
• Taiwan 886
• Tajikistan 992
• Tanzania 255
• Thailand 66
• Timor-Leste 670
• Togo 228
• Tokelau 690
• Tonga 676
• Tunisia 216
• Turkey 90
• Turkmenistan 993
• Turks & Caicos Islands 1-649
• Tuvalu 688
• U.S. Outlying Islands
• U.S. Virgin Islands 1-340
• UK 44
• US 1
• Uganda 256
• Ukraine 380
• United Arab Emirates 971
• Uruguay 598
• Uzbekistan 998
• Vanuatu 678
• Vatican City 39-06
• Venezuela 58
• Vietnam 84
• Wallis & Futuna 681
• Western Sahara 212
• Yemen 967
• Zambia 260
• Zimbabwe 263
• Less Than 6 Years
• 6 To 10 Years
• 11 To 16 Years
• Greater Than 16 Years

Methods of Rationalizing The Denominator

There are two common ways of rationalizing the denominator of a fraction depending on the type of expression present in the fraction.

These two ways are

• Using the conjugate of an irrational number
• Using an algebraic identity

Rationalize The Denominator Using The Conjugate of Irrational Number

A conjugate of an irrational expression or a similar irrational number is a similar irrational expression with an opposite sign. If the sign with an irrational number is positive then the sign with the conjugate will be negative and vice-versa.

For example, the conjugate of $3 + \sqrt{2}$ is $3 – \sqrt{2}$ and the conjugate of $3 – \sqrt{2}$ is $3 + \sqrt{2}$.

Note: To write the conjugate of an irrational number, you change the sign of an irrational part not that of the rational part.

For example, conjugate of $2 + \sqrt{3}$ is $2 – \sqrt{3}$ and not the $-2 + \sqrt{3}$. Similarly, $-2 – \sqrt{3}$ is not the conjugate of $2 + \sqrt{3}$.

In the process of rationalizing a denominator, the conjugate is the rationalizing factor. The process of rationalizing the denominator with its conjugate is as follows.

Step 1: Multiply both the denominator and numerator by a suitable conjugate that will remove the irrational number or irrational expression in the denominator

Step 2: Make sure that all the irrational numbers in the given fraction are in their simplified form

Step 3: Simplify the fraction further, if required

Examples

Let’s consider some examples to understand the process of rationalizing the denominator using the conjugate of an irrational number.

Ex 1: Rationalize the denominator of the fraction $\frac {1}{3 + \sqrt{2}}$.

Conjugate of $3 + \sqrt{2}$ is $3 – \sqrt{2}$.

Therefore, the rationalizing factor is $3 – \sqrt{2}$.

Multiplying the numerator and the denominator of the fraction by the rationalizing factor (RF).

$\frac {1}{3 + \sqrt{2}} \times \frac {3 – \sqrt{2}}{3 – \sqrt{2}} = \frac {1 \times \left(3 – \sqrt{2}\right)}{\left(3 + \sqrt{2}\right)\left(3 – \sqrt{2}\right)}$

$= \frac {3 – \sqrt{2}}{3^{2} – (\sqrt{2})^{2}}$

Note: ${\left(3 + \sqrt{2}\right)\left(3 – \sqrt{2}\right)}$ is of the form $\left(a + b\right)\left(a – b\right)$, where $a = 3$ and $b = \sqrt{2}$ and $\left(a + b\right)\left(a – b\right) = a^{2} – b^{2}$.

Therefore, $\frac {1 \times \left(3 – \sqrt{2}\right)}{\left(3 + \sqrt{2}\right)\left(3 – \sqrt{2}\right)} = \frac {3 – \sqrt{2}}{3^{2} – \left(\sqrt{2}\right)^{2}} = \frac {3 – \sqrt{2}}{9 – 2} = \frac {3 – \sqrt{2}}{7} = \frac {3}{7} – \frac {\sqrt{2}}{7}$.

Therefore, after rationalizing the denominator, $\frac {1}{3 + \sqrt{2}}$ becomes $\frac {3}{7} – \frac {\sqrt{2}}{7}$.

Ex 2: Rationalize the denominator of the fraction $\frac {2}{\sqrt{2} + 1}$

Conjugate of the denominator $\sqrt{2} + 1$ is $\sqrt{2} – 1$.

Multiplying the numerator and the denominator by $\sqrt{2} – 1$.

$\frac {2}{\sqrt{2} + 1} = \frac {2}{\sqrt{2} + 1} \times \frac {\sqrt{2} – 1}{\sqrt{2} – 1}$

$= \frac {2 \times \left(\sqrt{2} – 1\right)}{\left(\sqrt{2} + 1\right) \times \left(\sqrt{2} – 1\right)}$

$= \frac {2\sqrt {2} – 2 }{\left(\sqrt{2}\right)^{2} – 1^{2}}$

$= \frac {2 \sqrt{2} – 2}{2 – 1} = \frac {2 \sqrt{2} – 2}{1} = 2 \sqrt{2} – 2$

Therefore, after rationalizing the denominator $\frac {2}{\sqrt{2} + 1}$ becomes $2 \sqrt{2} – 2$.

Rationalize The Denominator Using Algebraic Identities

Using algebraic identities is another way of rationalizing the denominator of a fraction. The algebraic identity used in the process is $a^{2} – b^{2} = \left(a – b\right)\left(a + b\right)$.

• For rationalizing $\left(\sqrt{a} – \sqrt{b} \right)$, the rationalizing factor is $\left(\sqrt{a} + \sqrt{b} \right)$
• For rationalizing $\left(\sqrt{a} + \sqrt{b} \right)$, the rationalizing factor is $\left(\sqrt{a} – \sqrt{b} \right)$

Examples

Let’s consider some examples to understand the process of rationalizing the denominator using algebraic identities.

Ex 1: Rationalize the denominator of the fraction $\frac {1}{\sqrt {3} – \sqrt{2}}$.

Rationalizing factor (RF) is $\sqrt {3} + \sqrt{2}$.

Multiplying the numerator and the denominator of the fraction by $\sqrt {3} + \sqrt{2}$.

$\frac {1}{\sqrt {3} – \sqrt{2}} = \frac {1}{\sqrt {3} – \sqrt{2}} \times \frac {\sqrt {3} + \sqrt{2}}{\sqrt {3} + \sqrt{2}}$

$= \frac {1\times \left(\sqrt {3} + \sqrt{2} \right)}{\left(\sqrt {3} – \sqrt{2}\right)\left(\sqrt {3} + \sqrt{2}\right)}$

$= \frac {\sqrt{3} + \sqrt{2}}{\left(\sqrt{3}\right)^{2} – \left(\sqrt{2}\right)^{2}}$

$= \frac {\sqrt{3} + \sqrt{2}}{3 – 2} = \frac {\sqrt{3} + \sqrt{2}}{1} = \sqrt{3} + \sqrt{2}$

Ex 2: Rationalize the denominator of the fraction $\frac {5}{\sqrt {11} + \sqrt{7}}$.

Rationalizing factor (RF) is $\sqrt {11} – \sqrt{7}$.

Multiplying the numerator and the denominator by $\sqrt {11} – \sqrt{7}$.

$\frac {5}{\sqrt {11} + \sqrt{7}} \times \frac {\sqrt {11} – \sqrt{7}}{\sqrt {11} – \sqrt{7}}$

$= \frac {5 \times \left(\sqrt{11} – \sqrt{7}\right)}{\left(\sqrt{11} + \sqrt{7}\right)\left(\sqrt{11} – \sqrt{7}\right)}$

$= \frac {5\sqrt{11} – 5\sqrt{7}}{11 – 7} = \frac {5\sqrt{11} – 5\sqrt{7}}{4} = \frac {5\sqrt{11}}{4} – \frac {5\sqrt{7}}{4}$

Rationalize The Denominator With $3$ or More Terms

The same procedure that was followed to rationalize the denominator with $2$ terms is used but with a little variation.

Let’s consider a denominator that three terms: $a + b + c$.

First of all, rationalize the denominator with two terms, i.e., $a + b$ by multiplying with its conjugate $a – b$.

Now, the term $a + b$ reduces to a single term and then take $c$ to repeat the same process.

Note: An expression with three terms $a + b + c$ can be written as $\left(a + b\right) + c$. Now, according to the identity $\left(a – b\right)\left(a + b\right) = a^{2} – b^{2}$, $\left(\left(a + b\right) – c\right)\left(\left(a + b\right) + c\right) = \left(a + b\right)^{2} – c^{2}$. Therefore, the conjugate of $a + b + c$ is $\left(\left(a + b\right) – c\right)$.

Examples

Let’s consider an example to understand the procedure.

Ex 1: Rationalize the denominator of the fraction $\frac {1}{1 + \sqrt{2} + \sqrt{3}}$.

$\frac {1}{1 + \sqrt{2} + \sqrt{3}}$ can be written as $\frac {1}{\left(1 + \sqrt{2}\right) + \sqrt{3}}$.

Therefore, the rationalizing factor will be $\left(1 + \sqrt{2}\right) – \sqrt{3}$.

Multiplying the numerator and the denominator with $\left(1 + \sqrt{2}\right) – \sqrt{3}$

$\frac {1}{\left(1 + \sqrt{2}\right) + \sqrt{3}} \times \frac {\left(1 + \sqrt{2}\right) – \sqrt{3}}{\left(1 + \sqrt{2}\right) – \sqrt{3}}$

$= \frac {1 \times \left(\left(1 + \sqrt{2}\right) – \sqrt{3}\right)}{\left(1 + \sqrt{2}\right) + \sqrt{3} \times \left(\left(1 + \sqrt{2}\right) – \sqrt{3}\right)}$

$= \frac {1 + \sqrt{2} – \sqrt{3}}{\left(1 + \sqrt{2}\right)^{2} – \left(\sqrt{3}\right)^{2}}$

$= \frac {1 + \sqrt{2} – \sqrt{3}}{1 + 2\sqrt{2} + 2 – 3 }$

$= \frac {1 + \sqrt{2} – \sqrt{3}}{2\sqrt{2}}$

The denominator still contains an irrational number $\sqrt{2}$, which can be removed by multiplying the numerator and the denominator by its rationalizing factor, i.e., $\sqrt{2}$.

$\frac {1 + \sqrt{2} – \sqrt{3}}{2\sqrt{2}} \times \frac {\sqrt{2}}{\sqrt{2}}$

$= \frac {\sqrt{2} + 2 – \sqrt{6}}{4}$

Therefore, $\frac {1}{1 + \sqrt{2} + \sqrt{3}}$ after rationalizing the denominator becomes $\frac {\sqrt{2} + 2 – \sqrt{6}}{4}$.

Conclusion

Rationalizing the denominator is the process of making the irrational denominator of a fraction rational so that the fraction can be reduced to its simplest form.

Practice Questions

1. Write the rationalizing factor of
• $\sqrt{5}$
• $-\sqrt{7}$
• $2 + \sqrt{5}$
• $1 – \sqrt{7}$
• $-3 + \sqrt{2}$
• $-2 – \sqrt{2}$
2. Rationalize the denominator of the following fractions
• $\frac {4}{\sqrt{11}}$
• $-\frac {3}{\sqrt{3}}$
• $\frac {1}{1 + \sqrt{2}}$
• $\frac {3}{3 – \sqrt{2}}$
• $\frac {9}{-5 + \sqrt{3}}$
• $\frac {2}{-2 – \sqrt{5}}$
• $\frac {1}{1 + \sqrt{3} + \sqrt{5}}$
• $\frac {3}{2 – \sqrt{2} + \sqrt{3}}$
• $\frac {2}{2 + \sqrt{7} – \sqrt{5}}$

FAQs

What does it mean to rationalize the denominator and simplify?

Rationalizing the denominator means the process of removing an irrational number from the denominator of a fraction. Once the irrational number or expression is removed from the denominator, the fraction can be reduced to its lowest form which is called simplifying the fraction.

What is the formula for rationalizing?

To rationalize the denominator, we multiply the denominator by the conjugate of the irrational number or irrational expression.

The conjugate of an irrational expression is an irrational expression with an opposite sign.

Why do we need to rationalize the denominator?

Denominator of a fraction is rationalized so that it can be reduced to its simplest/lowest form.

What is the rationalizing factor?

A rationalizing factor is a number or an expression that is multiplied by an irrational number or an irrational expression to convert it to a rational number or rational expression.

What is the rationalizing factor of $\sqrt{8}$?

Rationalizing factor of $\sqrt{8}$ is $\sqrt{2}$, since $\sqrt{8} \times \sqrt{2} = \sqrt{8 \times 2} = \sqrt{16} = 4$.